Problem: Simplify the following expression: $t = \dfrac{12z^3 - 96z^2}{-96z^2}$ You can assume $z \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $12z^3 - 96z^2 = (2\cdot2\cdot3 \cdot z \cdot z \cdot z) - (2\cdot2\cdot2\cdot2\cdot2\cdot3 \cdot z \cdot z)$ The denominator can be factored: $-96z^2 = - (2\cdot2\cdot2\cdot2\cdot2\cdot3 \cdot z \cdot z)$ The greatest common factor of all the terms is $12z^2$ Factoring out $12z^2$ gives us: $t = \dfrac{(12z^2)(z - 8)}{(12z^2)(-8)}$ Dividing both the numerator and denominator by $12z^2$ gives: $t = \dfrac{z - 8}{-8}$